## Abstract

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian 1-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra and a collection, , of invariant functions on , we give a formula for a Lagrangian multiform describing the commuting flows for on a coadjoint orbit in

. We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on

. The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.

. We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on

. The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.

Original language | English |
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Article number | 34 |

Number of pages | 52 |

Journal | Letters in Mathematical Physics |

Volume | 114 |

DOIs | |

Publication status | Published - 19 Feb 2024 |

## Keywords

- Lagrangian multiforms
- Integrable systems
- Classical r-matrix